Stephan Baier

• PhD
• Professor (Full) at Ramakrishna Mission Vivekananda University, Belur Math

For any given positive definite binary quadratic form (PBQF), we prove that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by this PBQF and satisfying $||\alpha n||<n^{-3/7+\varepsilon}$ for any fixed but arbitrarily small $\varepsilon>0$.

We improve the large sieve inequality with k t h k^{\mathrm {th}} -power moduli, for all k ≥ 4 k\ge 4 . Our method relates these inequalities to a variant of Waring’s problem with restricted k t h k^{\mathrm {th}} -powers. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two k t h k^{\ma...

Matomäki proved that if α∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in {\mathbb {R}}$$\end{document} is irrational, then there are infinitely many primes...

We prove that for every irrational number $\alpha$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form $p=\left[n^c\right]$ with $n\in \mathbb{N}$ such that $||\alpha p||<p^{-\theta}$.

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $\lambda_1x_1^2+\cdots +\lambda_nx_n^2\equiv \lambda_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $\lambda_1,...,\lambda_{n+1}$ are integer coefficients such that $(\lambda_1\cdots \lambda _{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ a...

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke $L$-functions.

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his c...

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley...

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his c...

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/4-\varepsilon}$ with radical rad$(s)\le x^{9/40}$. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for g...

Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. To be precise, their condition is $M\ge D...

Let p>5 be a fixed prime and assume that α1,α2,α3 are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form α1x12+α2x22+α3x32≡0modq with q=pn, where max{|x1|,|x2|,|x3|}≤N and (x1x2x3,p)=1. (In fact, we consider a smoothed version of this problem.) If α1,α2,α3 are fixed and n→∞, we establish an asymptotic formu...

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has s...

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary q...

The distribution of \(\alpha p\) modulo one, where p runs over the rational primes and \(\alpha \) is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents \(\nu >0\) one can establish the infinitude of primes p satisfying \(||\alpha p||\le p^{-\nu }\). The latest record in this regard is Kaisa Matomäki’...

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $p\alpha$ when $\alpha$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the dista...

We prove a lower and an upper bound for the large sieve with square moduli in function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in [3] and [6]. Our lower bound in the function field setting contradicts an upper bound obtained in [4]. Indeed, in [5] we pointed out an error in [4].

In 2020, Roger Baker proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose car...

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent $1/4$ obtained by the first and third named authors for imaginary quadratic number fields~\cite{BT} and by th...

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-\nu}$. The latest record in this regard is Kaisa Matom\"aki's lan...

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot h...

Let n be a positive integer and f (x) := x 2 n + 1. In this paper, we study orders of primes dividing products of the form P m,n := f (1)f (2) · · · f (m). We prove that if m > max{10 12 , 4 n+1 }, then there exists a prime divisor p of P m,n such that ord p (P m,n) ≤ n · 2 n−1. For n = 2, we establish that for every positive integer m, there exist...

We point out an error in [2].

In [11], the second and third-named authors established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in [3] by the first and second-named authors. In this context, a Central Limit Theorem was established only under a strong hypothesis going beyo...

In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound...

The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a str...

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to I. M. Vinogradov, we obtain that the inequality $...

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every posi...

We prove a lower bound for the large sieve with square moduli.

We prove a lower bound for the large sieve with square moduli.

We establish a general large sieve inequality with sparse sets $\mathcal{S}$ of moduli in the Gaussian integers which are in a sense well-distributed in arithmetic progressions. This extends earlier work of S. Baier on the large sieve with sparse sets of moduli. We then use this result to obtain large sieve bounds for the cases when $\mathcal{S}$ c...

Let E be an elliptic curve over \( {\mathbb Q}\). Let p be a prime of good reduction for E. Then, for a prime \( p \ne \ell \), the Frobenius automorphism associated with p (unique up to conjugation) acts on the \( \ell \)-adic Tate module of E. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is ind...

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.

We establish a large sieve inequality for power moduli in ℤ[i], extending earlier work by Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in ℤ. Our method starts with a version of the large sieve for ℝ2. We convert the resulting counting problem back into one for ℤ[i] which we then attack using W...

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements...

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements...

We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$.

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $\ell \not= p$, the Frobenius automorphism (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism is defined over $\mathbb{Z}$ and is independent of $\ell$. Its s...

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $p\gamma$, $p$ running over the primes and $\gamma$ being a fixed irrational, for Gaussian primes.

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. The novelty lies in establishing power savings instead of power of logarithm saving, as obtained in earlier works. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of t...

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained by W.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and the fist-named author in (arXiv:math/0608318) for the first and second moments, but this comes at the cost of larger ranges of ave...

We establish a Polya-Vinogradov-type bound for finite periodic multipicative characters on the Gaussian integers.

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.

We investigate the distribution of $p\theta$ modulo 1, where $\theta$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 4...

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 4...

We prove a large sieve inequality for square norm moduli in Z[i].

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free Δ = D/16, where D is the discriminant, by the author and Browning [Inhomogeneous cubic congruences and rational points on Del Pezzo surfaces, J. Reine Angew. Math. 680 (2013) 6...

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.

Let F be an algebraically closed field of positive characteristic p. The third author and Will Turner gave an explicit description of the extension algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit basis. We examine this basis and use it to give upper and lower bounds for the growth behaviour of the dimensions of Ext-gr...

We give an alternative proof of a recent result by T. D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ...

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.

The dynamical Mertens ’ theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynam...

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.

In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.

We investigate the density of integer solutions to certain binary inhomogeneous quadratic congruences and use this information to detect almost primes on a singular del Pezzo surface of degree 6.

We investigate the first and second moments of shifted convolutions of the generalised divisor function d3(n).

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.

We investigate various mean value problems involving order 3 primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large sieve-type result for order 3 (and 6) Dirichlet chara...

In this paper, we develop a conditional subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$. Comment: A subtle mistake in Lemma 3 has been found and it does not appear that it can be easily fixed.

We obtain a new average results on the Lang-Trotter conjecture on elliptic curves.

Let $\lambda(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $\lambda(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form...

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Diric...

Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato–Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r, where r...

We establish a result on the large sieve with square moduli. These bounds improve recent results by S. Baier [S. Baier, On the large sieve with sparse sets of moduli, J. Ramanujan Math. Soc. 21 (2006) 279–295] and L. Zhao [L. Zhao, Large sieve inequality for characters to square moduli, Acta Arith. 112 (3) (2004) 297–308].

Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where...

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse–Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consi...

We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves.

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consi...

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+\e...

Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying p λ-θ < p-(1-λ)/2+ ε (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {p λ-θ} < p -min{(2-λ)/6,(14-9λ)/32}+ε. Thus we obtai...

We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.

In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\leq p^{5/9+\epsilon}$.

We deal with the distribution of the fractional parts of $p^{\lambda}$, $p$ running over the prime numbers and $\lambda$ being a fixed real number lying in the interval $(0,1)$. Roughly speaking, we study the following question: Given a real $\theta$, how small may $\delta>0$ be choosen if we suppose that the number of primes $p\le N$ satisfying ${...

We prove that the conditions $\lambda<5/19$ and $L\le T^{1/2}$ in Theorems 3 and 4 of our recent paper "On the $p^{\lambda}$ problem" can be omitted.

We establish a result on the large sieve with square moduli. These bounds impro ve recent results by S. Baier(math.NT/0512228) and L. Zhao(math.NT/0508125).

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when S consists of squares. These bounds improve a recent result by L. Zhao...

Suppose that a > 2. We prove that the number of positive integers q ≦ Q such that there exists a primitive character χ modulo q with χ (n) = 1 for all n ≦ (log Q)a is O(Q 1/(1-a)+ε).

We establish a large sieve bound for expressions of the form $$\sum\limits_{r=1}^R \left\vert \sum\limits_{M < n\le M+N} a_ne\left(\alpha_rf(n)\right)\right\vert^2,$$ where $f(x)=\alpha x^2+\beta x+\theta\in \mathbb{R}[x]$ is a quadratic polynomial with $\alpha>0$ and $\beta\ge 0$. We also consider the case when $f(x)=x^d$ with $d\in \mathbb{N}$, $...

Suppose that a > 1. By using a method of Linnik employing his Large Sieve one may derive the following result. The number of primitive Dirichlet characters χ with conductor ≤Q such that χ(n) = 1 for all n ≤ (log Q)a is O(Q2/a+ε). We improve the exponent 2/a for a > 2 by using a refined version by Heath-Brown of the Halasz–Montgomery ‘Large Values m...

We give a short alternative proof using Heath-Brown's square sieve of a bound of the author for the large sieve with square moduli.

In this paper we aim to generalize the results in Baier and Zhao and develop a general formula for large sieve with characters to powerful moduli that will be an improvement to the result of Zhao.

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao.

In several papers A. Balog, G. Harman and the author studied the distribution of p λ (mod 1), where λ is a given real number lying in the interval (0,1) and p runs over the prime numbers. One of the main questions in these papers can be formulated in the following way: Let a real θ be given. For what fixed positive real numbers τ is it possible to...